A phase fitted FinDiff process for DifEquns in quantum chemistry
Original Paper
First Online:
- 16 Downloads
Abstract
A newly FinDiff (\(=\) Finite Difference) process is developed for the productive application on the DifEquns (\(=\) Differential Equations) in Chemistry.
Keywords
Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schrödinger equationMathematics Subject Classification
65L05Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest or other ethical conflicts concerning this paper.
References
- 1.M.M. Chawla, S.R. Sharma, Families of 5th order Nyström methods for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ X, } \text{ Y })\) and intervals of periodicity. Computing 26(3), 247–256 (1981)CrossRefGoogle Scholar
- 2.J.M. Franco, M. Palacios, High-order P-stable multistep methods. J. Comput. Appl. Math. 30, 1–10 (1990)CrossRefGoogle Scholar
- 3.J.D. Lambert, Numerical Methods for Ordinary Differential Systems, the Initial Value Problem (Wiley, New York, 1991), pp. 104–107Google Scholar
- 4.E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)CrossRefGoogle Scholar
- 5.M.M. Chawla, S.R. Sharma, Intervals of periodicity and absolute stability of explicit Nyström methods for Y”=F(X, Y). Bit 21(4), 455–464 (1981)CrossRefGoogle Scholar
- 6.M.M. Chawla, Unconditionally stable Noumerov-type methods for 2nd order differential-equations. Bit 23(4), 541–542 (1983)CrossRefGoogle Scholar
- 7.
- 8.M.M. Chawla, P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of 2nd order periodic initial-value problems. J. Comput. Appl. Math. 11(3), 277–281 (1984)CrossRefGoogle Scholar
- 9.M.M. Chawla, Numerov made explicit has better stability. Bit 24(1), 117–118 (1984)CrossRefGoogle Scholar
- 10.M.M. Chawla, P.S. Rao, High-accuracy P-stable methods for \(\text{ Y }^{\prime \prime } = \text{ F }(\text{ T,Y })\), Ima J. Numer. Anal. 5(2), 215–220 (1985) and M.M Chawla, Correction, Ima J. Numer. Anal. 6(2) 252–252 (1986)Google Scholar
- 11.T. Lyche, Chebyshevian multistep methods for ordinary differential equations. Numer. Math. 19, 65–75 (1972)CrossRefGoogle Scholar
- 12.T.E. Simos, P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)CrossRefGoogle Scholar
- 13.R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)CrossRefGoogle Scholar
- 14.J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)CrossRefGoogle Scholar
- 15.M.M. Chawla, A new class of explicit 2-step 4th order methods for \(\text{ Y }^{\prime \prime } = \text{ F }(\text{ T, } \text{ Y })\) with extended intervals of periodicity. J. Comput. Appl. Math. 14(3), 467–470 (1986)CrossRefGoogle Scholar
- 16.M.M. Chawla, B. Neta, Families of 2-step 4th-order P-stable methods for 2nd-order differential-equations. J. Comput. Appl. Math. 15(2), 213–223 (1986)CrossRefGoogle Scholar
- 17.M.M. Chawla, P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of 2nd-order periodic initial-value problems. 2. Explicit method. J. Comput. Appl. Math. 15(3), 329–337 (1986)CrossRefGoogle Scholar
- 18.M.M. Chawla, P.S. Rao, B. Neta, 2-Step 4th-order P-stable methods with phase-lag of order 6 for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ T, } \text{ Y })\). J. Comput. Appl. Math. 16(2), 233–236 (1986)CrossRefGoogle Scholar
- 19.M.M. Chawla, P.S. Rao, An explicit 6th-order method with phase-lag of order 8 for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ T, } \text{ Y })\). J. Comput. Appl. Math. 17(3), 365–368 (1987)CrossRefGoogle Scholar
- 20.M.M. Chawla, M.A. Al-Zanaidi, Non-dissipative extended one-step methods for oscillatory problems. Int. J. Comput. Math. 69(1–2), 85–100 (1998)CrossRefGoogle Scholar
- 21.M.M. Chawla, M.A. Al-Zanaidi, A two-stage fourth-order “almost” P-stable method for oscillatory problems. J. Comput. Appl. Math. 89(1), 115–118 (1998)CrossRefGoogle Scholar
- 22.M.M. Chawla, M.A. Al-Zanaidi, S.S. Al-Ghonaim, Singly-implicit stabilized extended one-step methods for second-order initial-value problems with oscillating solutions. Math. Comput. Model. 29(2), 63–72 (1999)CrossRefGoogle Scholar
- 23.J.P. Coleman, Numerical-methods for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ X, } \text{ Y })\) via rational-approximations for the cosine. Ima J. Numer. Anal. 9(2), 145–165 (1989)CrossRefGoogle Scholar
- 24.J.P. Coleman, A.S. Booth, Analysis of a family of Chebyshev methods for \(\text{ Y }^{\prime \prime } = \text{ F }(\text{ X, } \text{ Y })\). J. Comput. Appl. Math. 44(1), 95–114 (1992)CrossRefGoogle Scholar
- 25.J.P. Coleman, L.G. Ixaru, P-stability and exponential-fitting methods for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ X, } \text{ Y })\). Ima J. Numer. Anal. 16(2), 179–199 (1996)CrossRefGoogle Scholar
- 26.J.P. Coleman, S.C. Duxbury, Mixed collocation methods for \(\text{ Y } ^{\prime \prime } = \text{ F }(\text{ X, } \text{ Y })\). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)CrossRefGoogle Scholar
- 27.L.G. Ixaru, S. Berceanu, Coleman method maximally adapted to the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 11–20 (1987)CrossRefGoogle Scholar
- 28.L.G. Ixaru, The Numerov method and singular potentials. J. Comput. Phys. 72(1), 270–274 (1987)CrossRefGoogle Scholar
- 29.L.G. Ixaru, M. Rizea, Numerov method maximally adapted to the Schrödinger-equation. J. Comput. Phys. 73(2), 306–324 (1987)CrossRefGoogle Scholar
- 30.L.G. Ixaru, H. De Meyer, G. Vanden Berghe, M. Van Daele, Expfit4—a Fortran program for the numerical solution of systems of nonlinear second-order initial-value problems. Comput. Phys. Commun. 100(1–2), 71–80 (1997)CrossRefGoogle Scholar
- 31.L.G. Ixaru, G. Vanden Berghe, H. De Meyer, M. Van Daele, Four-step exponential-fitted methods for nonlinear physical problems. Comput. Phys. Commun. 100(1–2), 56–70 (1997)CrossRefGoogle Scholar
- 32.L.G. Ixaru, M. Rizea, Four step methods for \(\text{ Y }^{\prime \prime }=\text{ F }(\text{ X,Y })\). J. Comput. Appl. Math. 79(1), 87–99 (1997)CrossRefGoogle Scholar
- 33.M. Van Daele, G. Vanden Berghe, H. De Meyer, L.G. Ixaru, Exponential-fitted four-step methods for \(\text{ Y } ^{\prime \prime }=\text{ F }(\text{ X,Y })\). Int. J. Comput. Math. 66(3–4), 299–309 (1998)CrossRefGoogle Scholar
- 34.L.G. Ixaru, B. Paternoster, A conditionally P-stable fourth-order exponential-fitting method for \(\text{ Y } ^{\prime \prime } = \text{ F }(\text{ X, } \text{ Y })\). J. Comput. Appl. Math. 106(1), 87–98 (1999)CrossRefGoogle Scholar
- 35.L.G. Ixaru, Numerical operations on oscillatory functions. Comput. Chem. 25(1), 39–53 (2001)PubMedCrossRefPubMedCentralGoogle Scholar
- 36.L.G. Ixaru, G. Vanden Berghe, H. De Meyer, Exponentially fitted variable two-step BDF algorithm for first order odes. Comput. Phys. Commun. 150(2), 116–128 (2003)CrossRefGoogle Scholar
- 37.M.A. Medvedev, T.E. Simos, A three-stages multistep teeming in phase algorithm for computational problems in chemistry. J. Math. Chem. 57(6), 1598–1617 (2019)CrossRefGoogle Scholar
- 38.M.A. Medvedeva, T.E. Simos, An accomplished phase FD process for DEs in chemistry, J. Math. Chem. (in press)Google Scholar
- 39.Y.-Y. MA, C.-L. Lin, T.E. Simos, An integrated in phase FD procedure for diffeqns in chemical problems, J. Math. Chem. (to appear)Google Scholar
- 40.M. Xu, T.E. Simos, A multistage two-step fraught in phase scheme for problems in mathematical chemistry. J. Math. Chem. 57(7), 1710–1731 (2019)CrossRefGoogle Scholar
- 41.J. Lv, T.E. Simos, A Runge-Kutta type crowded in phase algorithm for quantum chemistry problems. J. Math. Chem. 57(8), 1983–2006 (2019)CrossRefGoogle Scholar
- 42.F. Hui, T.E. Simos, A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)CrossRefGoogle Scholar
- 43.L.G. Ixaru, M. Rizea, Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)CrossRefGoogle Scholar
- 44.L.G. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)Google Scholar
- 45.L.G. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)CrossRefGoogle Scholar
- 46.J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge–Kutta–Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)CrossRefGoogle Scholar
- 47.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
- 48.G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)CrossRefGoogle Scholar
- 49.A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)CrossRefGoogle Scholar
- 50.M.M. Chawla, P.S. Rao, An Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems II explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)CrossRefGoogle Scholar
- 51.M.M. Chawla, P.S. Rao, An explicit sixth-order method with phase-lag of order eight for \(y^{\prime \prime }=f(t, y)\). J. Comput. Appl. Math. 17, 363–368 (1987)Google Scholar
- 52.M. Rizea, Exponential fitting method for the time-dependent Schrödinger equation. J. Math. Chem. 48(1), 55–65 (2010)CrossRefGoogle Scholar
- 53.A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)CrossRefGoogle Scholar
- 54.A.D. Raptis, J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 36, 113–119 (1985)CrossRefGoogle Scholar
- 55.A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)CrossRefGoogle Scholar
- 56.R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. R. Soc. Ser. A 274, 427–442 (1963)CrossRefGoogle Scholar
- 57.R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)CrossRefGoogle Scholar
- 58.M. Rizea, V. Ledoux, M. Van Daele, G. Vanden Berghe, N. Carjan, Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission. Comput. Phys. Commun. 179(7), 466–478 (2008)CrossRefGoogle Scholar
- 59.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)CrossRefGoogle Scholar
- 60.K. Mu, T.E. Simos, A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53, 1239–1256 (2015)CrossRefGoogle Scholar
- 61.L.G. Ixaru, M. Rizea, G. Vanden Berghe, H. De Meyer, Weights of the exponential fitting multistep algorithms for first-order odes. J. Comput. Appl. Math. 132(1), 83–93 (2001)CrossRefGoogle Scholar
- 62.A.D. Raptis, J.R. Cash, Exponential and bessel fitting methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 95–103 (1987)CrossRefGoogle Scholar
- 63.C.D. Papageorgiou, A.D. Raptis, A method for the solution of the Schrödinger-equation. Comput. Phys. Commun. 43(3), 325–328 (1987)CrossRefGoogle Scholar
- 64.Z. Zhou, T.E. Simos, A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54, 442–465 (2016)CrossRefGoogle Scholar
- 65.A.D. Raptis, Exponential multisteo methods for ordinary differential equations. Bull. Greek Math. Soc. 25, 113–126 (1984)Google Scholar
- 66.H. Ning, T.E. Simos, A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J. Math. Chem. 53(6), 1295–1312 (2015)CrossRefGoogle Scholar
- 67.Z. Wang, T.E. Simos, An economical eighth-order method for the approximation of the solution of the Schrödinger equation. J. Math. Chem. 55, 717–733 (2017)CrossRefGoogle Scholar
- 68.K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties, J. Math. Chem. (in press)Google Scholar
- 69.J.R. Cash, A.D. Raptis, A high-order method for the numerical-integration of the one-dimensional Schrödinger-equation. Comput. Phys. Commun. 33(4), 299–304 (1984)CrossRefGoogle Scholar
- 70.A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control. Comput. Phys. Commun. 28(4), 427–431 (1983)CrossRefGoogle Scholar
- 71.A.D. Raptis, 2-Step methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 28(4), 373–378 (1982)Google Scholar
- 72.A.D. Raptis, On the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 24(1), 1–4 (1981)CrossRefGoogle Scholar
- 73.A.D. Raptis, Exponential-fitting methods for the numerical-integration of the 4th-order differential-equation \(\text{ Y }^{iv}+\text{ F.Y }=\text{ G }\). Computing 24(2–3), 241–250 (1980)CrossRefGoogle Scholar
- 74.H. Van De Vyver, A symplectic exponentially fitted modified Runge–Kutta–Nyström method for the numerical integration of orbital problems. New Astron. 10(4), 261–269 (2005)CrossRefGoogle Scholar
- 75.H. Van De Vyver, On the generation of P-stable exponentially fitted Runge–Kutta–Nyström methods by exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 188(2), 309–318 (2006)CrossRefGoogle Scholar
- 76.M. Van Daele, G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations. Numer. Algorithms 44(2), 115–131 (2007)CrossRefGoogle Scholar
- 77.M. Van Daele, G. Vanden Berghe, P-stable exponentially-fitted Obrechkoff methods of arbitrary order for second-order differential equations. Numer. Algorithms 46(4), 333–350 (2007)CrossRefGoogle Scholar
- 78.Y. Fang, W. Xinyuan, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions. Appl. Numer. Math. 58(3), 341–351 (2008)CrossRefGoogle Scholar
- 79.G. Vanden Berghe, M. Van Daele, Exponentially-fitted Obrechkoff methods for second-order differential equations. Appl. Numer. Math. 59(3–4), 815–829 (2009)CrossRefGoogle Scholar
- 80.D. Hollevoet, M. Van Daele, G. Vanden Berghe, The optimal exponentially-fitted numerov method for solving two-point boundary value problems. J. Comput. Appl. Math. 230(1), 260–269 (2009)CrossRefGoogle Scholar
- 81.J.M. Franco, L. Rández, Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs. Appl. Math. Comput. 273, 493–505 (2016)Google Scholar
- 82.J.M. Franco, Y. Khiar, L. Rández, Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems. Appl. Math. Comput. 252, 45–57 (2015)Google Scholar
- 83.J.M. Franco, I. Gomez, L. Rández, Optimization of explicit two-step hybrid methods for solving orbital and oscillatory problems. Comput. Phys. Commun. 185(10), 2527–2537 (2014)CrossRefGoogle Scholar
- 84.J.M. Franco, I. Gomez, Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs. Appl. Math. Comput. 232, 643–657 (2014)Google Scholar
- 85.J.M. Franco, I. Gomez, Symplectic explicit methods of Runge–Kutta–Nyström type for solving perturbed oscillators. J. Comput. Appl. Math. 260, 482–493 (2014)CrossRefGoogle Scholar
- 86.J.M. Franco, I. Gomez, Some procedures for the construction of high-order exponentially fitted Runge–Kutta–Nyström methods of explicit type. Comput. Phys. Commun. 184(4), 1310–1321 (2013)CrossRefGoogle Scholar
- 87.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On some new low storage implementations of time advancing Runge–Kutta methods. J. Comput. Appl. Math. 236(15), 3665–3675 (2012)CrossRefGoogle Scholar
- 88.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Symmetric and symplectic exponentially fitted Runge–Kutta methods of high order. Comput. Phys. Commun. 181(12), 2044–2056 (2010)CrossRefGoogle Scholar
- 89.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages. BIT Numer. Math. 50(1), 3–21 (2010)CrossRefGoogle Scholar
- 90.J.M. Franco, I. Gomez, Accuracy and linear stability of RKN methods for solving second-order stiff problems. Appl. Numer. Math. 59(5), 959–975 (2009)CrossRefGoogle Scholar
- 91.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type. J. Comput. Appl. Math. 223(1), 387–398 (2009)CrossRefGoogle Scholar
- 92.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Structure preservation of exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 218(2), 421–434 (2008)CrossRefGoogle Scholar
- 93.M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted modified Runge–Kutta methods of Gauss type. Comput. Phys. Commun. 178(10), 732–744 (2008)CrossRefGoogle Scholar
- 94.J.M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177(6), 479–492 (2007)CrossRefGoogle Scholar
- 95.J.M. Franco, New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56(8), 1040–1053 (2006)CrossRefGoogle Scholar
- 96.J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)CrossRefGoogle Scholar
- 97.J.M. Franco, Stability of explicit ARKN methods for perturbed oscillators. J. Comput. Appl. Math. 173, 389–396 (2005)CrossRefGoogle Scholar
- 98.X.Y. Wu, X. You, J.Y. Li, Note on derivation of order conditions for ARKN methods for perturbed oscillators. Comput. Phys. Commun. 180, 1545–1549 (2009)CrossRefGoogle Scholar
- 99.A. Tocino, J. Vigo-Aguiar, Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods. Math. Comput. Model. 42, 873–876 (2005)CrossRefGoogle Scholar
- 100.L. Brugnano, F. Iavernaro, D. Trigiante, Hamiltonian boundary value methods (energy preserving discrete line integral methods). JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, 17–37 (2010)Google Scholar
- 101.F. Iavernaro, D. Trigiante, High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4, 87–101 (2009)Google Scholar
- 102.A. Konguetsof, A generator of families of two-step numerical methods with free parameters and minimal phase-lag. J. Math. Chem. 55(9), 1808–1832 (2017)CrossRefGoogle Scholar
- 103.A. Konguetsof, A hybrid method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation. J. Math. Chem. 49(7), 1330–1356 (2011)CrossRefGoogle Scholar
- 104.H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems. Int. J. Mod. Phys. C 17(5), 663–675 (2006)CrossRefGoogle Scholar
- 105.H. Van de Vyver, An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53(9), 1339–1348 (2007)CrossRefGoogle Scholar
- 106.Y. Fang, W. Xinyuan, A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)Google Scholar
- 107.B. Neta, P-stable high-order super-implicit and Obrechkoff methods for periodic initial value problems. Comput. Math. Appl. 54(1), 117–126 (2007)CrossRefGoogle Scholar
- 108.H. Van de Vyver, Phase-fitted and amplification-fitted two-step hybrid methods for y” = f (x, y). J. Comput. Appl. Math. 209(1), 33–53 (2007)CrossRefGoogle Scholar
- 109.H. Van de Vyver, Efficient one-step methods for the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 711–732 (2008)Google Scholar
- 110.J. Martín-Vaquero, J. Vigo-Aguiar, Exponential fitted Gauss, Radau and Lobatto methods of low order. Numer. Algorithms 48(4), 327–346 (2008)CrossRefGoogle Scholar
- 111.A. Konguetsof, A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)CrossRefGoogle Scholar
- 112.F.A. Hendi, P-stable higher derivative methods with minimal phase-lag for solving second order differential equations, J. Appl. Math. 2011 Article ID 407151 (2011)Google Scholar
- 113.H. Van de Vyver, Comparison of some special optimized fourth-order Runge–Kutta methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 166(2), 109–122 (2005)CrossRefGoogle Scholar
- 114.Z. Wang, D. Zhao, Y. Dai, W. Dongmei, An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 461(2058), 1639–1658 (2005)CrossRefGoogle Scholar
- 115.M. Van Daele, G. Vanden Berghe, H. De Meyer, Properties and implementation of \(\rho \)-adams methods based on mixed-type interpolation. Comput. Math. Appl. 30(10), 37–54 (1995)CrossRefGoogle Scholar
- 116.J. Vigo-Aguiar, L.M. Quintales, A parallel ODE solver adapted to oscillatory problems. J. Supercomput. 19(2), 163–171 (2001)CrossRefGoogle Scholar
- 117.Z. Wang, Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation. Comput. Phys. Commun. 174(2), 109–118 (2006)CrossRefGoogle Scholar
- 118.Z. Wang, Trigonometrically-fitted method for a periodic initial value problem with two frequencies. Comput. Phys. Commun. 175(4), 241–249 (2006)CrossRefGoogle Scholar
- 119.J. Vigo-Aguiar, J.M. Ferrandiz, A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems. SIAM J. Numer. Anal. 35(4), 1684–1708 (1998)CrossRefGoogle Scholar
- 120.J. Vigo-Aguiar, H. Ramos, Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations. J. Comput. Appl. Math. 158(1), 187–211 (2003)CrossRefGoogle Scholar
- 121.J. Vigo-Aguiar, S. Natesan, A parallel boundary value technique for singularly perturbed two-point boundary value problems. J. Supercomput. 27(2), 195–206 (2004)CrossRefGoogle Scholar
- 122.C. Tang, H. Yan, H. Zhang, W.R. Li, The various order explicit multistep exponential fitting for systems of ordinary differential equations. J. Comput. Appl. Math. 169(1), 171–182 (2004)CrossRefGoogle Scholar
- 123.C. Tang, H. Yan, H. Zhang, Z. Chen, M. Liu, G. Zhang, The arbitrary order implicit multistep schemes of exponential fitting and their applications. J. Comput. Appl. Math. 173(1), 155–168 (2005)CrossRefGoogle Scholar
- 124.H. Van de Vyver, Frequency evaluation for exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 184(2), 442–463 (2005)CrossRefGoogle Scholar
- 125.J.P. Coleman, L.G. Ixaru, Truncation errors in exponential fitting for oscillatory problems. SIAM J. Numer. Anal. 44(4), 1441–1465 (2006)CrossRefGoogle Scholar
- 126.J. Martín-Vaquero, J. Vigo-Aguiar, Adapted BDF algorithms: higher-order methods and their stability. J. Sci. Comput. 32(2), 287–313 (2007)CrossRefGoogle Scholar
- 127.J. Vigo-Aguiar, J. Martín-Vaquero, H. Ramos, Exponential fitting BDF-Runge–Kutta algorithms. Comput. Phys. Commun. 178(1), 15–34 (2008)CrossRefGoogle Scholar
- 128.B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday. Comput. Phys. Commun. 183(12), 2499–2512 (2012)CrossRefGoogle Scholar
- 129.Z. Wang, Obrechkoff one-step method fitted with Fourier spectrum for undamped Duffing equation. Comput. Phys. Commun. 175(11–12), 692–699 (2006)CrossRefGoogle Scholar
- 130.C. Wang, Z. Wang, A P-stable eighteenth-order six-step method for periodic initial value problems. Int. J. Mod. Phys. C 18(3), 419–431 (2007)CrossRefGoogle Scholar
- 131.J. Chen, Z. Wang, H. Shao, H. Hao, Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff method. Comput. Phys. Commun. 179(7), 486–491 (2008)CrossRefGoogle Scholar
- 132.H. Shao, Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 180(1), 1–7 (2009)CrossRefGoogle Scholar
- 133.H. Shao, Z. Wang, Numerical solutions of the time-dependent Schrödinger equation: reduction of the error due to space discretization, Phys. Rev. E 79(5) Article ID 056705 (2009)Google Scholar
- 134.Z. Wang, H. Shao, A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. Comput. Phys. Commun. 180(6), 842–849 (2009)CrossRefGoogle Scholar
- 135.T.E. Simos, Exponentially fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation and related problems. Comput. Mater. Sci. 18, 315–332 (2000)CrossRefGoogle Scholar
- 136.C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)Google Scholar
- 137.F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)Google Scholar
- 138.A.R. Leach, Molecular Modelling—Principles and Applications (Pearson, Essex, 2001)Google Scholar
- 139.P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford Univ. Press, Oxford, 2011)Google Scholar
- 140.V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Y.A. Khakhalev, A.N. Zolotov, Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations, in AIP Conference Proceedings, vol. 738 (2016), p. 480004Google Scholar
- 141.V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow, in AIP Conference Proceedings, vol. 1648 (2015), p. 850033Google Scholar
- 142.N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika 1, 49–53 (1998)Google Scholar
- 143.V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Modeling and development of cooling technology of turbine engine blades. Int. Rev. Mech. Eng. 9(4), 331–335 (2015)Google Scholar
- 144.S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar
Copyright information
© Springer Nature Switzerland AG 2019